The research of (G,w)-Chaos and G-Lipschitz shadowing property

Abstract

In this paper, we introduce the concepts of (G,w)− Chaos and G− Lipschitz shadowing property. We study the dynamical properties of (G,w)− Chaos in the inverse limit space under group action. In addition, we study the dynamical properties of G− Lipschitz shadowing property respectively under topological G− conjugate and iterative systems. The following conclusions are obtained. (1) Let (Xf,G¯, d¯,σ) be the inverse limit space of (X,G,d,f) under group action. If the self-map f is (G,w)− chaotic, the shift map σ is (G,w)− chaotic; (2) Let (X,d) be a metric G− space and f be topologically G− conjugate to g. Then the map f has G− Lipschitz shadowing property if and only if the map g has G− Lipschitz shadowing property. (3) Let (X,d) be a metric G− space and f be an equivariant Lipschitz map from X to X. Then for any positive integer k⩾2, the map f has the G− Lipschitz shadowing property if and only if the iterative map fk has the G− Lipschitz shadowing property. These results enrich the theory of topological G− conjugate, iterative system and the inverse limit space under group action.

Keywords

• inverse limit space
• topological g− conjugate
• iterative map
• (g，w)− chaos
• g− lipschitz shadowing property